Chapter 3 Compoud iterest 1
Simple iterest ad compoud amout formula Formula for compoud amout iterest is: S P ( 1 Where : S: the amout at compoud iterest P: the pricipal i: the rate per coversio period : the umber of coversio periods the factor factor. ( 1 called the accumulatio Note: 1 I may busiess trasactios, the iterest is computed aually, semiaually, quarterly, mothly, daily, or at some other time iterval. The importat rate is the iterest rate per coversio period, which is desigated by the symbol i.
3 The quoted aual rate is called the omial rate, ad is idicated by the symbol m. the equatio relatig j, m, i is i=j/m, or j= i. m ("jim", the symbol j (m meas a omial rate coverted m times a year. 4 Whe o coversio period is stated i problem assume that the iterest is compouded aually. Fidig Example 1 How may semiaual coversio periods are there from Jue 1, 1989, to December 1, 1994? Solutio: Year Moth day 1994 1 1-1989 -6-1 5 years 6 moths 0 days 3
= (5 x + 6/6 = 10 + 1 = 11 after subtractio we multiply the umber of years by the period per year ad divide the umber of moths by the moths i a period, ad add the results to get the umber of periods. Example How may quarterly coversio period are from November 15, 1990, to August 15, 1999. Solutio: = ( 8 X 4 + 9/3 = 3 + 3 = 35. Example 3 Fid the computed a mout of $5 ivested at 6% coverted quarterly for 5 years. 4
Solutio : S P ( 1 0.06 5(1 4 5 1.3469 54 $33.67 Example 4 A pricipal of $1000 is deposited at 6% for 10 years. What will be the computed amout ad the compoud iterest if the iterest is compouded aually, semiaually, quarterly, ad mothly? Solutio : - iterest computed aually: S P ( 1 1000(1 $1790.85 0.06 10 Compoud iterest= S - P =$1790.85 - $1000=$790.85. - iterest computed semiaually: 5
S P ( 1 1000(1 $1806.11 0.06 10 Compoud iterest= S - P =$1806.11 - $1000=$806.11. - iterest computed quarterly : S P ( 1 1000(1 $1814.0 0.06 4 104 Compoud iterest= S - P =$1814.0 - $1000=$814.0. - iterest computed mothly : S P ( 1 1000(1 $1819.40 0.06 1 101 Compoud iterest= S - P =$1819.40 - $1000=$819.40 6
Example 5 A perso aged 60 put $10000 i a deferred a accout payig 8% coverted quarterly. The a accout is to mature i 5 years. Fid the amout at that time. Solutio: Substitutig P=10000, i=0.08/4=0.0, =5*4=0 S P ( 1 10000(1 0.08 4 54 $14859.47 Example 6 A bak pays 7.8% compouded quarterly o savigs accouts. A woma puts $5000 ito such a accout o July 1, 1990. fid amout i the accout o Jauary 1, 1995. Solutio : 7
N = 4 * 4 + 6/3 = 18 periods S P ( 1 5000(1 $7078.48 0.078 4 18 Example 7 A depositor plaed to leave $000 i savig ad loa associatio payig 5% compouded semiaually for a period of 5 years. At the ed of.5 years the depositor had to withdraw $1000. what amout will be i the accout at the ed of origial 5-year period? Solutio: First we compute the a mout after.5 years 8
S P ( 1 000(1 0.05.5 $6.8 After withdrawal of $1000 the fial balace is $ 6.8 - $ 1000 = $16.8 The amout at the ext.5 years S P ( 1 16.8(1 0.05.5 $148.76 Alterate solutio Usig the equatio of value 9
000(1 0.05 10 1000( 1 0.05 5 x x = $ 148.76 low of orgaic growth example 1 durig the period 1970 1980, the populatio of a city icreased at rate of a bout 3% a year. If the populatio i 1980 was 300 000, what is the predicted populatio i 1990? Solutio: Substitutig P=300 000, i=0.03, =10 S P ( 1 Example : 300000(1 403175 0.03 10 11
Durig the period 1970-1975 the earig per share of Oklahoma atural Gas Compay commo stock icreased at about 9% a year compouded. The earigs per share for 1975 was $.9. assumig that the same rate of icrease cotiues, predict the earigs per share for 1990. Solutio: Substitutig P=.9, i=0.09, =15 S P ( 1.9(1 0.09 $10.63 15 Example 3: Durig the period 1970-1980 the populatio of a city icreased 8%. if the populatio was 500 000 i 1980, what is the estimated populatio for 11
000 assumig that the same rate of growth cotiues? Substitutig P=500 000, i=0.08, = S P ( 1 500000(1 0.08 58300 H.W :page 119 Exercise 3a (all Example ( daily compoudig Compute the amout of 1-factor for 5% coverted daily for days for 360- ad 365 years. Solutio : For 360 day year, we fid that 1
S ( 1 j 0.05 (1 360 360 $1.000777 For 365 day year, we fid that S ( 1 j 0.05 (1 365 365 $1.0007399 H.W :page 16 Exercise 3b (all 13
14
Iterest for a part of a period. whe derivig the compoud iterest formula, we assume that the time would be a iteger umber of coversio periods, whe there is a part of period, the usual practice is to allow simple 15
iterest for this time o the computed amout at the ed of the last whole period. Example At 7% compouded semiaually, $000 will amout to how much i 3 years ad 5 moths? The total time i this case is 6 period ad 5 moths left over. Solutio : The amout after 6 period is 16
S P ( 1 000(1 $458.51 0.07 6 The simple iterest for the remaiig 5 moths is I = P. r t = 458.51 * 0.07 *(5/1= $71.71 So the amout at the ed if 3 years ad 5 moths is 458.51 + 71.71 = $530. Amout at chagig rates Example : A Pricipal of $900 ears 6% coverted quarterly for 4 years ad the 7% coverted semiaually for more years. Fid the fial amout. Solutio : 17
First we fid the amout at the ed of 4 years ( 16 period S P ( 1 0.06 900(1 4 $114.09 Secod we fid the amout at the ext of yeas years ( 4 period S P ( 1 114.09(1 $1310.57 H.W :page 130 Exercise 3c (all 16 0.07 4 Preset value at compoud iterest P S ( 1 Where : P: the pricipal or preset value 18
S: the amout due i the future i: the rate per period : the umber of periods Example :1 Fid the preset value of $5000 due i 4 years if moey is worth 8% compouded semiaually. Solutio: Substitutig S =5000, i=0.08, =8 S 5000 P $3653.45 ( 1 0.08 8 (1 Example : Fid the preset value of $7500 due i 4 years if moey is worth 14% compouded mothly. Solutio: Substitutig S =7500, i=0.14/1, =4*1=48 19
S 7500 P $497.98 ( 1 0.14 48 (1 1 Example :3 How much must be ivested i a accout payig 8.4% compouded mothly i order to accumulate to $15000 i 5 years. Solutio: Substitutig S =$15000, i=0.084/1 = 0.007, =5*1=60 S 15000 P $9870.13 ( 1 0.084 60 (1 1 Example :4 A ote with a matuary value of $1000 is due i3 years ad 8 moths. What is its preset value at 6% compouded semiaually? Solutio: 1
Number of iteger period = 7, ad the part of last period = moths P P S (1 r. t (1 1000 0.06 S 999.09 ( 1 0.06 7 (1 Example 5 : 1 $999.09 $805.04 O August 5, 1985, Mr. Kae loaded Ms. Hill $000 at 1% coverted semiaually. Mr. Hill gave Mr Ka a ot promisig to repay the loa with accumulated iterest i 6 years. O February 5, 1989, Mr Ka sold the ote to a buyer, who charge a iterest rate of 16% 1
coverted semiaually for discoutig. How much did Mr. Ma get? Solutio : Step 1: S P ( 1 000(1 $404.39 0.1 1 Step :
P S 404.39 ( 1 0.16 5 (1 $738.95 Example 6: A perso ca buy a piece of property for $4500 cash or $000 dow ad $3000 i 3 years. If the perso has moey earig 6% coverted semiaually, which is the better purchase pla ad how much ow? Solutio: We get the preset value of $3000 due i 3 years at 6% compouded semiaually. S 3000 P $51.45 ( 1 0.06 6 (1 3
Addig this amout to the $000 dow paymet makes the preset value of this paymet pla $451.45. By payig $4500 cash, the buyer saves $1.45 ow. Example 7: A piece of property ca be purchased for $ 850 cash or for $3000 i 1 moths. Which is the better pla for the buyer if moey is worth 7% compouded quarterly? Fid the cash ( preset value equivalet of the savig made by adoptig the better pla. Solutio : Put the focal date ow: P S 3000 ( 1 0.07 4 (1 4 $798.88 4
It is better to pay later. Ad the cash equivalet of the savig is = 850 798.88 = $ 51.1 H.W :page 138 Exercise 3d, ad 3e page 143 (all Fidig the rate Example If $500 amouts to $700 i 5 years with iterest compouded quarterly, what is the rate of iterest? Solutio : 5
S P ( 1 0 700 500(1 0 (1 1.4 0 1 i 1.4 1.01697 i 0.01697 j m. i 4 0.01697 0.0679 6.79% Ecoomic aalysis: Example: Per capita persoal icome i the uited state icreased from $841 i 1980 to $13157 i 1987. what was the aual compouded rate of retur? Solutio: 6
S P ( 1 13157 (1 841(1 7 1.56 7 1 i i 7 1.56 1.0658 0.0658 6.58% H.W :page 151 Exercise 3f (all Fidig the time : Example 1: How log will it take $00 to amout to $350 at 7% compouded semiaually? Solutio: 7
S P ( 1 350 00(1 (1 0.035 log(1 0.035 0.07 1.75 log(1.75 log( 1.035 log( 1.75 log(1.75 16.673 log(1.035 16.673 t 8.1336 years H.W :page 175 Exercise 3g (all Equatio of value Example 1: A perso owes $0000 due i 1 year ad $0000 due i years. The leder agrees to the settlemet of both obligatios with a cash 8
paymet. assume that the rate used equal 10% compouded semiaually. determie the size of the cash paymets. Solutio: Equatio of value used as: x 0000 0.10 (1 18140.56 $481.66 30000 0.10 (1 4681.07 4 Example : 9
A perso owes $50000 due ow. The leder agrees to settle this obligatio with equal paymets i 1 ad years, respectively, fid the size of the paymets if the settlemet is based o 9%. Solutio: x (1 0.9 1 x 50000( 1 0.09.09 x $59405 x $843.44 Example 3: 31
A piece of property is sold for $50000. the buyer pays $0000 cash, ad sigs a o-iterestbearig ot for $10000 due i 1 year, ad secod a o-iterest-bearig ot for $10000 due i years. If the seller charges 10% compouded aually, what a o-iterest-bearig ot due i 3 years will pay off the debt? Solutio: Equatio of value as follows: 30000(10.10 10000(1 39930 1100 11000 x x $16830 3 0.10 10000(1 0.10 1 x 31
Example 4: A perso owes $0000 due i 3 years with iterest at 10% compouded quarterly, ad $10000 due i 5 years with iterest at 8%. If moey is worth 9% what sigle paymet 6 years hece will be equivalet to the origial obligatios?. Solutio: First we obtai the maturity values of the debts: 3
The amout of %0000 after 3 years = 0000(1 0.10 4 1 $6897.78 The amout of $10000 after 5 years = 10000(1 0.08 5 $14693.8 The equatio of value as: x 6897.78(1 0.09 34833.41 16015.68 $50849.09 3 14693.8(1 0.09 1 Example 5: O Jue 1, 1991, a perso obtaied a $5000 loa for which paymet of $1000 o the pricipal plus 6% iterest o the upaid balace will be made every 6 moths. The paymet schedule for this loa is give i the followig chart: 33
Paymet umber Paymet Paymet Total Balace Date o o paymet of loa iterest pricipal Jue 1, 1991 5000 1 Dec.1,1991 1300 300 * 1000 4000 Jue 1,199 140 40 1000 3000 3 Dec 1, 199 1180 180 1000 000 4 Jue 1, 1993 110 10 1000 1000 5 Dec 1,1993 1060 60 1000 0 * iterest at Dec. 1,1991 = 5000*0.06*1 O Jue 1, 199 the leder sells this cotract to a buyer how wats a yield of 16% coverted semiaually. Fid the sale price. Solutio : 34
Put the focal date at Jue 1, 199, the equatio value is: 1180 110 1060 x 0.16 1 0.16 0.16 (1 (1 (1 109.59 960. 841.46 $894.7 3 35
EXAMPLE 6. A head of household stipulates i a will that $30000 from the bequeathed estate is to be placed i a fud from which each of the three childre i a family is to receive a equal amout upo reached age 1. whe the head of the household dies, the childre are ages 19, 16, ad 14. if the fud is ivested at 85 coverted semiaually, how much does each receive? Solutio : 36
The equatio value as: x 300000 0.08 4 (1 (1 0.854804 x 0.675564 x x x 0.08 10 0.08 14 (1 0.5774754 x 30000 x $143.56 H.W :page 168, 177 Exercise 3h, 3i (all 37
Coefficiet. 38
Solutio: 39